REMARKS ABOUT ENERGY TRANSFER IN AN RC LADDER NETWORK
WOJCIECHMITKOWSKI∗
∗Department of Automatics, Academy of Mining and Metallurgy al. Mickiewicza 30/B-1, 30–059 Kraków, Poland
e-mail:wmi@ia.agh.edu.pl
The problem of energy transfer in an RC-ladder network is considered. Using the maximum principle, an algorithm for constructing optimal control is proposed, where the cost function is the energy delivered to the network. In the case considered, optimal control exists. Numerical simulations were performed using Matlab.
Keywords: optimal control, energy transfer, approximation of an RC-long line
1. Introduction
The problem of determining optimal controls is one of fundamental problems in control theory and its applica- tions (Athans and Falb, 1969; Bryson and Ho, 1972). An- alytic solutions exist only in particular examples. Below we will investigate such an example.
i(t) iw(t)
R1 TRANSMISSION
LINE RH y(t) )
(t u
Fig. 1. Scheme of energy transfer.
We consider an electric network shown in Fig. 1. The resistance of the voltage source R1 and the output resis- tance RH are given. Let
J (u) =
T
Z
0
u(t)i(t) dt, (1)
where J (u) is the energy delivered to the network and T is the time horizon. Let the output current be iw(0) = 0 and t ∈ [0, T ]. Assume that T and E are fixed and consider the following problem: Find uo such that
J (u) ≥ J (uo), ∀ u (2a) and
Z T 0
y(t)iw(t) dt = E, (2b)
where E is the energy producing heat on the resis- tance RH.
Remark 1. Consider the electric network shown in Fig. 2.
We have
J (u) = 1
R1+ R + RH Z T
0
u(t)2dt
E = RH
(R1+ R + RH)2
T
Z
0
u(t)2dt. (3)
i(t) iw(t) R
R1 TRANSMISSION
LINE RH y(t) )
(t u
Fig. 2. Electric resistance network.
The energy E producing heat on the resistance RH is given. From (3) we obtain many controls u satisfying
Z T 0
u(t)2dt = E(R1+ R + RH)2/RH,
J (u) = E
1 + R1+ R RH
. (4)
If u(t) = ¯u = const, then
¯
u = (R1+ R + RH)
r E
RHT.
Remark 2. Consider a homogeneous long electric RC transmission line, i.e. one where the parameters per the unit length (resistance r and capacity c) are constant and independent of the co-ordinate z. An infinitesimal part of the long line is described by the equation
rc∂x(t, z)
∂t =∂2x(t, z)
∂z2 , 0 ≤ t, 0 ≤ z ≤ l. (5)
Remark 3. Let z = ih, h = l/n, i = 0, 1, . . . , n and x(t, (2k − 1)h/2) = xk(t), k = 1, 2, . . . , n. We have
∂2x(t, z)
∂z2 ≈1 h
x(t, z + h) − x(t, z) h
− x(t, z) − x(t, z − h) h
for z = (2k − 1)h/2 and k = 1, 2, . . . , n. Then the RC transmission line can be approximated by the RC ladder network shown in Fig. 3, where R = rl and C = cl (Butkovskii, 1965, p. 314).
R 2/ n R /n R /n R 2/ n
R1 C /n C /n C /n RH u(t) x1(t) x2(t) xn(t) y(t) APPROXIMATION OF RC TRANSMISSION LINE
Fig. 3. RC ladder network.
2. Electric RC Ladder Network
Consider the electric RC ladder network shown in Fig. 3.
Its parameters R, R1, RH and C are known. The sys- tem shown in Fig. 3 can be described by the equation (Mitkowski, 1994; 1997; 2000):
˙
x(t) = Ax(t) + Bu(t),
x(t) =x1(t) x2(t) . . . xn(t)T
, (6)
y(t) = W x(t),
where A is the n × n real tridiagonal Jacobi matrix, A = [aij], aij= 0 for |i − j| > 1, aii= n2
RC, i = 2, 3, . . . , n − 1, a11= −(1 + r(R1))n2
RC,
ann= −(1 + r(RH))n2
RC, r(γ) = 2R 2nγ + R, ai,i−1= n2
RC, i = 2, 3, . . . , n, ai,i+1= n2
RC, i = 1, 2, 3, . . . , n − 1, B =n2r(R1)
RC e1, e1= [1 0 0 . . . 0 0]T, W =nr(RH)RH
R [0 0 . . . 0 1]. (7)
For fixed n the tridiagonal real Jacobi matrix A has only single real eigenvalues λi. The matrix A is diag- onalizable. The Jordan canonical form of A is J = diag (λ1, . . . , λn). From Gershgorin’s criterion and the fact that det A 6= 0, we have λi ∈ [−m, 0), where m = maxi(|ai,i−1| + |ai,i+1|). Thus (Mitkowski, 2000, p. 301) the system (6) is asymptotically stable.
3. Problem Formulation and Its Solution
Consider the system (6). Let x(0) = 0 and (cf. Eqn. (1)) the cost function be
J (u) = Z T
0
u(t)i(t) dt
= 2n
2nR1+ R Z T
0
u(t)[u(t) − x1(t)] dt, (8) where J (u) is the energy delivered to the electric RC- network, and T is the time horizon.
Optimal control problem: Let T and E be fixed. Find a control uo∈ Ud such that
J (u) ≥ J (uo), ∀ u ∈ Ud,
Ud= (
u : 1 RH
Z T 0
y(t)2dt
= n2RH (nRH+ R/2)2
Z T 0
xn(t)2dt = E )
, (9)
where E is the energy producing heat on the resistance RH (see Fig. 3) and Ud is the set of admissible controls.
Remark 4. The set Ud is non-empty. Indeed, examine, e.g. u(t) = const such that
1 RH
Z T 0
y(t)2dt = n2RH
(nRH+ R/2)2
T
Z
0
xn(t)2dt = E
(cf. (6) and (7) for x(0) = 0). Now, we consider the spaces Lp(0, T ), p ∈ [1, ∞) with the norms kf kp = [RT
0 |f (t)|pdt]1/p. From the Hölder inequal- ity (Musielak, 1976, p. 45; Luenberger, 1974, p. 58) we have RT
0 u(t)x1(t) dt ≤ kux1k1≤ kuk2kx1k2. The sys- tem (6) is asymptotically stable, controllable and observ- able (cf. (7); the pair (A, B) is controllable and (W, A) is observable). Consequently,
(R1+ R/2n)J (u) = kuk22− Z T
0
u(t)x1(t) dt
≥ kuk22− kuk2kx1k2≥ −kx1k22/4 (see (8) and (9)), for every u ∈ Ud the norm kx1k2 is finite and J (u) → ∞ as kuk → ∞. Thus there exists the optimal control uo, cf. (8). We can notice that J (u) = J (−u).
The Maximum Principle makes it possible to con- struct an algorithm for determining optimal control.
Defining new state variables
˙
xn+1(t) = xn(t)2, xn+1(0) = 0,
˙
xn+2(t) = u(t)[u(t) − x1(t)], xn+2(0) = 0, (10) we have
xn+1(T ) =(nRH+ R/2)2 n2RH
E,
xn+2(T ) =2nR1+ R 2n J (u).
Let ˜x(t) = [x(t)T xn+1(t) xn+2(t)]T and ˜ψ(t) = [ψ(t)T ψn+1(t) ψn+2(t)]T. Then we obtain the Hamilto- nian in the form
H ˜ψ(t), ˜x(t), u(t)
= ψ(t)TAx(t) + Bu(t) + ψn+1(t)xn(t)2 + ψn+2(t)u(t)u(t) − x1(t). (11) In this case ψn+1(t) = −ρ = const, ψn+2(t) = −1, ψ(T ) = 0 and (the adjoint system)
ψ(t) = −A˙ Tψ(t) − b,
bT=u(t) 0 0 . . . 0 − 2ρxn(t), (12) where ψ is the adjoint function. Using the Maximum Principle (Pontriagin et al., 1983, Górecki, 1993, p. 393), from (11) we get
u(t) =1
2BTψ(t) + x1(t)
=1 2
2n2
(2nR1+ R)Cψ1(t) + x1(t)
. (13)
The control (13) depends on the real number ρ and is called the extremal control. The optimal control uo can exist only among the extremal controls (13).
From (6), (12) and (13), we obtain the canonical sys- tem in the following form:
"
˙ x(t) ψ(t)˙
#
=
"
Z1 Z2
Z3 Z4
# "
x(t) ψ(t)
# ,
x(0) = 0, ψ(T ) = 0, (14) where the matrices Zi (depending on ρ) are given by the closed-loop system (6), (12) and (13). Let
Z =
"
Z1 Z2
Z3 Z4
#
, eZt =
"
Φ1(t) Φ2(t) Φ3(t) Φ4(t)
# . (15) Then from (14) and (15) we have
x(t) = Φ2(t)ψ(0), ψ(t) = Φ4(t)ψ(0). (16) If E 6= 0, then x(t) 6= 0, cf. (9). Thus from (16) we get ψ(0) 6= 0. Since ψ(T ) = 0, cf. (14), from (16) we have
det Φ4(T ) = 0. (17) The idea of the control algorithm:
• Determine the parameter ρ using Eqn. (17).
• From (9) and (16) calculate ψ(0).
• From (13) and (16) determine u(t) =1
2BTψ(t) + x1(t)
=1
2BTΦ4(t) + eT1Φ2(t)ψ(0), (18) where e1= [1 0 0 . . . 0 0]T ∈ Rn.
4. RC Ladder Network with n=1
A very interesting case corresponds to n = 1. This is because closed-form formulae for the optimal trajecto- ries can be obtained, in particular for the optimal control uo(t), as well as a closed-form formula for the cost func- tion J (uo).
Now we consider an RC ladder network shown in Fig. 3 with n = 1. In this case we obtain the following parameters (see (6)):
A = − R1+ RH+ R C(R1+ R/2)(RH+ R/2),
B = 1
C(R1+ R/2), (19)
W = RH
RH+ R/2,
and in (15) we have
Z1= − 2R1+ RH+ 3R/2 2C(R1+ R/2)(RH+ R/2),
Z2= 1
2C2(R1+ R/2)2, (20) Z3= 2ρ − 1
2, Z4= −Z1.
Remark 5. (Górecki, 1993, p. 394, 584; Korytowski, 2001). The matrix Z for n = 1, cf. (15), has eigenvalues λ1= λ and λ2= −λ, where λ =pZ12+ Z2Z3. Assume that rank Z > 0. If Z3 = −Z12/Z2, then λ = 0 (only one eigenvector corresponds to λ = 0, be- cause rank Z > 0) and Φ4(t) = 1−tZ1. In this case (17) cannot be exploited, because Z1< 0 and t > 0.
If λ 6= 0, then closed-form formulae for the elements Φ2(t) and Φ4(t) of the matrix eZt (see (15)) for n = 1 are given by
Φ2(t) =Z2(eλt− e−λt)
2λ ,
Φ4(t) =(λ − Z1)eλt+ (λ + Z1)e−λt
2λ , (21)
λ = q
Z12+ Z2Z3, where the Zi’s are given in (20).
From (17) and (21) we have e2λt= (Z1+ λ)/(Z1− λ). If λ is real, λ > 0 and t > 0, then e2λt 6= (Z1+ λ)/(Z1− λ).
Now, if Z12+ Z2Z3< 0, then λ1= λ, λ2= −λ, λ = jω, j2= −1, ω =
q
|Z12+ Z2Z3| (22) and consequently, from (21), we obtain
Φ2(t) =Z2
ω sin ωt, Φ4(t) = cos ωt−Z1
ω sin ωt. (23) Thus from (23) we can notice that (17) holds for the ap- propriate $t.
We can notice that Z12+ Z2Z3< 0 if and only if ρ < −2R1+ R
2RH+ R
2R1+ R 2RH+ R + 1
= ρd. (24)
From (17) we conclude that Φ4(T ) = 0. Because in this case Φ4(t) is given by (23), we have the following equation:
tan z = −Kz, K = − 1
Z1T, z = ωT. (25)
It has many (positive) solutions:
zi∈ π/2 + (i − 1)π, π + (i − 1)π,
i = 1, 2, 3, . . . . (26) For every zi there exists
ρi= − (2R1+ R)Czi
2T
2
+ ρd, (27) cf. (20) and (22), where ρd is given in (24).
From (9) and (16) we have Z T
0
x1(t)2dt = Z T
0
Φ2(t)2dtψ1(0)2
=(RH+ R/2)2 RH
E. (28)
In this case (n = 1) the number ψ1(0) is dependent on zi, cf. (26) (or ρi, cf. (27)). From (28) we have
ψ1(0) = ±(RH+ R/2) zi
Z2T s
2E(1 + K2zi2) RHT (1 + K + K2zi2), K = − 1
Z1T, (29)
where the Zi’s are given in (20).
It is easy to show that, using elementary operations (cf. (8) and (16)), we have
J (u) = 1 (R1+ R/2)
Z T 0
u(t)u(t) − x1(t) dt
= ψ1(0)2 4(R1+ R/2)
× Z T
0
4
(2R1+ R)2C2Φ4(t)2−Φ2(t)2
dt. (30) Consequently, from (30), (23) and (29) we obtain
J u(ρi) = E
1 + R1+ R RH
+ (RH+ R/2)2(R1+ R/2)C2 RHT2 z2i
. (31)
Remark 6. We can notice (cf. (31)), that J (u(ρ1)) <
J (u(ρi)), ∀ i, where ρi is given by (27) and zi is given by (25) and (26).
Remark 7. From (9) we get
kx1k22= (RH+ R/2)2 RH
E.
Thus from Remark 4 we have (R1+ R/2)J (u) = kuk22−
Z T 0
u(t)x1(t) dt
≥ kuk22− kuk2kx1k2, (32) and consequently
(R1+ R/2)J (u) ≥ kuk22− kuk2p ˜RE ≥ − ˜RE/4, R =˜ (RH+ R/2)2
RH
. (33)
Since the function J (u) is continuous and (33) holds, there exists the optimal control uo, cf. (8). One can notice that J (u) = J (−u).
Using Remarks 6 and 7, we obtain optimal control (for ρ = ρ1) in the following form, cf. (18):
uo(t) =1 2
2
(2R1+ R)C
cos ωt −Z1
ω sin ωt
+Z2
ω sin ωt
ψ1(0), ω = z1/T, (34) where ψ1(0) is given by (29). The optimal trajectories are given by the following equalities:
x1(t) = ψ1(0)Z2
ω sin ωt, i(t) = u(t) − x1(t) R1+ R/2 . (35) Example 1. Let R1 = 1, R = 1, RH = 2, C = 1, T = 0.5, E = 10. Then for n = 1 we have K = 2.7272, cf. (25), Z1 = −0.7333, Z2 = 0.2222, z1 = 1.7746, ρ1 = −29.3013, ω = 3.5491, ψ1(0) = 169.3551 and J (uo) = 610.4439. The optimal control uo(t), ‘×−’, the optimal electric current i(t), ‘+−’, the function ψ1(t)
‘◦−’, and the optimal trajectory x1(t), ‘∗−’, are shown in Fig. 4.
5. RC Ladder Network with n=2
Consider the electric RC ladder network shown in Fig. 3 with n = 2. The parameters R, R1, RH and C are known. Equations (6) and (7) describe the system. In this case optimal control can be determined by numerical calculations.
Example 2. Let R1 = 1, R = 1, RH = 2, C = 1, T = 0.5 and E = 10. Then for n = 2 the parameters of the system are given by (7). The function ρ 7→ det Φ4(T ) is shown in Fig. 5. In this case det Φ4(T ) = 0 for ρ =
−43.757, ψ1(0) = 90.3034, ψ2(0) = −1.2256ψ1(0) and J (uo) = 609.7385. The optimal control uo(t),
‘×−’, the optimal electric current i(t), ‘+−’, the func- tion ψ1(t), ‘◦−’, ψ2(t), ‘∗−’ and the optimal trajectories x1(t), ‘·−’ and x2(t), ‘: −’ are shown in Fig. 6.
Fig. 4. Optimal trajectories for n = 1.
Fig. 5. Function ρ 7→ det Φ4(T ).
Fig. 6. Optimal trajectories for n = 2.
6. Concluding Remarks
In applications, optimal control problems are of paramount importance. Unfortunately, only in few exam- ples we can find closed-form formulae for optimal control.
In this paper such an example was studied. The resulting two-point boundary-value problem (14), (9) was analyti- cally solved (for n = 1). For large n we have to solve this problem numerically.
RC ladder networks constitute a kind of approxima- tions to RC-long lines (see Remarks 2 and 3). Probably, the results presented in this paper can be applied to dis- tributed systems. A very important problem is the transfer of an energy quantum in a given time with the simultane- ous minimization of the energy delivered to the system.
For example, it can be used in microelectronics, biology and engineering. Generally, the problem of energy mini- mization is very important.
Acknowledgment
This work was supported by the KBN-AGH Contract No. 11 11 120 230.
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Received: 22 December 2002 Revised: 2 April 2003
